are you behind a proxy? either that or you are having DNS issues resolving packages.linuxmint

anyway when you solve those connection problems prepare yourself because you have a huge update to perform (looking at the kernel you are using - and assuming it's the newest you have installed - the last UP you did was 3, we had 5 last month)

"konnte nicht aufgelöst werden" >> "couldn't resolve". Obviously a DNS issue. Maybe the Linux Mint team changed stuff and DNS servers didn't refresh the records (in this case it will be fixed in 24-48 hours at most). Or proxy config missing / proxy blacklisting. Actually both APTs work (debian / packages <- if you can get these with your web browser: system proxy to configure).